Vertex equitable labeling of signed graphs

A signed graph (or, in short, sigraph) S=(Su,σ) consists of an underlying graph Su:=G=(V,E) and a function σ:E(Su)→{+,−}, called the signature of S. Let S be a signed graph with p vertices and q edges and let A={0,1,2,…,⌈q2⌉}. A vertex labeling f:V(S)→A which is onto, is said to be a vertex equitable labeling of S if it induces a bijective edge labeling f⁎:E(S)→{1,2,…,m,−1,−2,…,−n} defined by f⁎(uv)=σ(uv)(f(u)+f(v)) such that |vf(a)−vf(b)|≤1, ∀a,b∈A, where vf(a) is the number of vertices with f(v) = a and m, n are number of positive and negative edges respectively in S. A signed graph S is said to be vertex equitable if it admits a vertex equitable labeling. In this paper, we initiate a vertex equitable labeling of signed graphs and study vertex equitable behavior of signed paths, signed stars and signed complete bipartite graphs K2,n.